Theorem r0cld 23703

Description: The analogue of the T₁ axiom (singletons are closed) for an R₀ space. In an R₀ space the set of all points topologically indistinguishable from 𝐴 is closed. (Contributed by Mario Carneiro, 25-Aug-2015.)

Hypothesis

Ref	Expression
kqval.2	⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})

Assertion

Ref	Expression
r0cld	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → {𝑧 ∈ 𝑋 ∣ ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝐴 ∈ 𝑜)} ∈ (Clsd‘𝐽))

Distinct variable groups:   𝑥,𝑜,𝑦,𝑧,𝐴   𝑜,𝐽,𝑥,𝑦,𝑧   𝑜,𝐹,𝑧   𝑜,𝑋,𝑥,𝑦,𝑧

Allowed substitution hints:   𝐹(𝑥,𝑦)

Proof of Theorem r0cld

Step	Hyp	Ref	Expression
1		kqval.2	. . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})
2	1	kqffn 23690	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
3	2	3ad2ant1 1134	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → 𝐹 Fn 𝑋)
4		fncnvima2 7013	. . . 4 ⊢ (𝐹 Fn 𝑋 → (◡𝐹 “ {(𝐹‘𝐴)}) = {𝑧 ∈ 𝑋 ∣ (𝐹‘𝑧) ∈ {(𝐹‘𝐴)}})
5	3, 4	syl 17	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → (◡𝐹 “ {(𝐹‘𝐴)}) = {𝑧 ∈ 𝑋 ∣ (𝐹‘𝑧) ∈ {(𝐹‘𝐴)}})
6		fvex 6853	. . . . . 6 ⊢ (𝐹‘𝑧) ∈ V
7	6	elsn 4582	. . . . 5 ⊢ ((𝐹‘𝑧) ∈ {(𝐹‘𝐴)} ↔ (𝐹‘𝑧) = (𝐹‘𝐴))
8		simpl1 1193	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
9		simpr 484	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ 𝑋)
10		simpl3 1195	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) → 𝐴 ∈ 𝑋)
11	1	kqfeq 23689	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝐹‘𝑧) = (𝐹‘𝐴) ↔ ∀𝑦 ∈ 𝐽 (𝑧 ∈ 𝑦 ↔ 𝐴 ∈ 𝑦)))
12		eleq2w 2820	. . . . . . . . 9 ⊢ (𝑦 = 𝑜 → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑜))
13		eleq2w 2820	. . . . . . . . 9 ⊢ (𝑦 = 𝑜 → (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ 𝑜))
14	12, 13	bibi12d 345	. . . . . . . 8 ⊢ (𝑦 = 𝑜 → ((𝑧 ∈ 𝑦 ↔ 𝐴 ∈ 𝑦) ↔ (𝑧 ∈ 𝑜 ↔ 𝐴 ∈ 𝑜)))
15	14	cbvralvw 3215	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝐽 (𝑧 ∈ 𝑦 ↔ 𝐴 ∈ 𝑦) ↔ ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝐴 ∈ 𝑜))
16	11, 15	bitrdi 287	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝐹‘𝑧) = (𝐹‘𝐴) ↔ ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝐴 ∈ 𝑜)))
17	8, 9, 10, 16	syl3anc 1374	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) → ((𝐹‘𝑧) = (𝐹‘𝐴) ↔ ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝐴 ∈ 𝑜)))
18	7, 17	bitrid 283	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) → ((𝐹‘𝑧) ∈ {(𝐹‘𝐴)} ↔ ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝐴 ∈ 𝑜)))
19	18	rabbidva 3395	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → {𝑧 ∈ 𝑋 ∣ (𝐹‘𝑧) ∈ {(𝐹‘𝐴)}} = {𝑧 ∈ 𝑋 ∣ ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝐴 ∈ 𝑜)})
20	5, 19	eqtrd 2771	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → (◡𝐹 “ {(𝐹‘𝐴)}) = {𝑧 ∈ 𝑋 ∣ ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝐴 ∈ 𝑜)})
21	1	kqid 23693	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
22	21	3ad2ant1 1134	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
23		simp2 1138	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → (KQ‘𝐽) ∈ Fre)
24		simp3 1139	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑋)
25		fnfvelrn 7032	. . . . . 6 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐹‘𝐴) ∈ ran 𝐹)
26	3, 24, 25	syl2anc 585	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → (𝐹‘𝐴) ∈ ran 𝐹)
27	1	kqtopon 23692	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
28	27	3ad2ant1 1134	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
29		toponuni 22879	. . . . . 6 ⊢ ((KQ‘𝐽) ∈ (TopOn‘ran 𝐹) → ran 𝐹 = ∪ (KQ‘𝐽))
30	28, 29	syl 17	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → ran 𝐹 = ∪ (KQ‘𝐽))
31	26, 30	eleqtrd 2838	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → (𝐹‘𝐴) ∈ ∪ (KQ‘𝐽))
32		eqid 2736	. . . . 5 ⊢ ∪ (KQ‘𝐽) = ∪ (KQ‘𝐽)
33	32	t1sncld 23291	. . . 4 ⊢ (((KQ‘𝐽) ∈ Fre ∧ (𝐹‘𝐴) ∈ ∪ (KQ‘𝐽)) → {(𝐹‘𝐴)} ∈ (Clsd‘(KQ‘𝐽)))
34	23, 31, 33	syl2anc 585	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → {(𝐹‘𝐴)} ∈ (Clsd‘(KQ‘𝐽)))
35		cnclima 23233	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn (KQ‘𝐽)) ∧ {(𝐹‘𝐴)} ∈ (Clsd‘(KQ‘𝐽))) → (◡𝐹 “ {(𝐹‘𝐴)}) ∈ (Clsd‘𝐽))
36	22, 34, 35	syl2anc 585	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → (◡𝐹 “ {(𝐹‘𝐴)}) ∈ (Clsd‘𝐽))
37	20, 36	eqeltrrd 2837	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → {𝑧 ∈ 𝑋 ∣ ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝐴 ∈ 𝑜)} ∈ (Clsd‘𝐽))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3051  {crab 3389  {csn 4567  ∪ cuni 4850   ↦ cmpt 5166  ^◡ccnv 5630  ran crn 5632   “ cima 5634   Fn wfn 6493  ‘cfv 6498  (class class class)co 7367  TopOnctopon 22875  Clsdccld 22981   Cn ccn 23189  Frect1 23272  KQckq 23658

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-map 8775  df-qtop 17471  df-top 22859  df-topon 22876  df-cld 22984  df-cn 23192  df-t1 23279  df-kq 23659

This theorem is referenced by:  nrmr0reg  23714